
There is a canonical isomorphism of Artin fans \(\mathcal {A}_{J_{g,n}^{log}}\cong \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\)  under which we have a factorization
\(\operatorname{trop}_{J_{g,n}^{log}}\colon J_{g,n}^{log}\xrightarrow{}\widetilde{\mathcal {J}}^{trop}_{g,n} \xrightarrow{}\mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}} \cong \mathcal {A}_{J_{g,n}^{log}},\) 
where the map \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}\)  (that we baptize the modular logarithmic tropicalization map) has a modular description (see Definition REF ) and it is strict, smooth and surjective.
Moreover, the diagram (REF ) commutes, via the suitable forgetful-stabilization morphisms, with the analogous diagram for \(\overline{\mathcal {M}}_{g,n}\)  established in [1]} and [2]}.

The two morphisms of logarithmic algebraic stacks
\({\overline{\mathcal {J}}_{g,n}&& J_{g,n}^{log} [ll]_{\Upsilon _{J_{g,n}^{log}}} [rr]^{\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}} && \widetilde{\mathcal {J}}^{trop}_{g,n}}\) 
are compatible with the toroidal stratification (REF ) of \(\overline{\mathcal {J}}_{g,n}\)  and the stratification (REF ) of \( \widetilde{\mathcal {J}}^{trop}_{g,n}\)  as an Artin fan.
Moreover, the diagram (REF ) is compatible with the restrictions to \(\overline{\mathcal {J}}_{g,n, d}\) , \(\overline{\mathcal {J}}_{g,n}^{\operatorname{spl}}\)  and \(\overline{\mathcal {J}}_{g,n}(\phi )\)  for any universal stability condition \(\phi \in V_{g,n}\) .


